Kinetic Energy Formula definition explanation

[Selina94]

Homework Statement

I'm trying to understand how this equation was came together, mainly, I hardly understand why velocity is squared, and why there's the constant of '1/2'.

Homework Equations

In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body depends on the mass of the body as well as its speed. The kinetic energy is equal to the mass multiplied by the square of the speed, multiplied by the constant 1/2. In formula form:

 

[hms.tech]

Work done on an object and one of the equation of motion for constant acceleration :

V^2 - U^2 = 2AD (A= acceleration vector)(D= Displacement vector)(the rest is self explanatory)
W=F.D

 

[vela]

Integrate the righthand side of
$$\int \vec{F}\cdot d\vec{r} = \int m\vec{a}\cdot d\vec{r}$$ using the chain rule to write $a_i = \frac{d v_i}{dt} = \frac{dv_i}{dx_i}\frac{dx_i}{dt}$.